Rationalizing the Denominator
A. What It Means to Rationalize the Denominator
In order that all of us doing math can compare answers, we agree upon a common
conversation, or set of rules, concerning the form of the answers. For instance, we
could easily agree that we would not leave an answer in the form of 3 + 4 but would
write 7 instead.
When the topic switches to that of radicals, those doing math have agreed that a
radical in simplest form will not (among other things) have a radical in the
denominator of a fraction. We will all change the form so there is no radical in the
denominator.
If a fraction contains a radical in the denominator such as √ which is an irrational
number, we need to make it NOT irrational, or rational. The process of changing its
form so it is no longer irrational is called rationalizing the denominator.
B. Different Cases of Rationalizing the Denominator
Case 1: A single square root in the denominator.
Example:
square root
√
one term
In a fraction, the numerator and denominator can be multiplied by the same value and still be an
equivalent fraction. For example,
for which both the top and bottom of the fraction were
multiplied by 3. We will use this idea in this example to create an equivalent fraction, but it will have a
rational denominator.
Procedure: Multiply the numerator and denominator of the fraction by the radical
that is currently in the denominator.
√
√
√
√
√
Look at what is happening here: The
denominator becomes √ which is a square root
that can be simplified. It simplifies to 3.
√
Now, the denominator is a rational number which was our goal.
Case 2: There is a single radical in the denominator, however, THE
INDEX IS GREATER THAN TWO. It might be a cube root or a fourth
root.
Example:
√
We need to multiply by something that will
allow us to completely simplify the cube
root. Currently there is one 3 and two x’s
under the radical: √
. So, we need
)and another x to be
two more 3’s (
able to completely simplify the cube root.
Rationalizing the Denominator
Procedure: Multiply the numerator and denominator by the radical that when
multiplied creates a perfect cube under the radical in the denominator.
√
√
=
√
So we can completely
simplify the cube root.
√
√
√
=
This now leaves a rational
denominator.
Case 3: There are TWO TERMS in the denominator.
Example:
Procedure:
√
We will multiply both numerator and denominator by the conjugate of the
denominator. The conjugate has the same two terms but with the opposite sign
between them.
The conjugate of
√ is
simplify to a rational number.
Be careful with the
multiplication in both the
numerator and denominator.
(
√ )
√ )( √ )
√
√
√ . Because of this, the denominator will
(
√
√
√
√
√
Notice that
the middle
terms cancel.
√
√
√
The denominator will now simplify
to a rational number.
√
Lastly, this fraction can be reduced. Since there are two terms in the numerator, we must be able to reduce
both of them with the denominator. In this case, we can:
√
=
√
=
√
This is our final answer.